How many flies are there?

Question

A group of spiders and flies: there are 1046 legs and 147 heads. How many flies are there

Answer 1

S = number of spiders at 8 legs and 1 head per spider

F = number of flies at 6 legs and 1 head per fly

 

147 heads ⇒ S + F = 147

 

1046 legs ⇒ 8S + 6F = 1046

 

Solve system of equations:

 

(i)  S + F = 147

(ii) 8S + 6F = 1046

 

From (i) get S = 147 - F and substitute into (ii) to find F:

 

8(147 - F) + 6F = 1046

1176 - 8F + 6F = 1046

2F = 130

F = 130/2 = 65

 

Substitute F = 65 into (i) and get S = 147 - 65 = 82

 

There are 65 flies and 82 spiders.

 

Test the answer:

 

Is 65 + 82 = 147 (yes)

Is 6(65) + 8(82) = 1046 (yes)

 

Answer 2

Let:

  s = number of spiders

  f = number of flies

 

Spiders and flies each have one head. So,

  "147 heads"    means    s + f = 147 

                           and   s = 147 - f  [eq1; eliminate s because we want to find f]

 

Spiders have 8 legs and flies have 6 legs.  So,

  "there are 1046 legs"     means 

                      8s + 6f = 1046    [eq2]

 

The math:

    8s + 6f = 1046              [eq2 again]

    8(147-f) + 6f = 1046      [substitute from eq1]

    1176 -8f + 6f = 1046        [distribute]

         -2f = -130             [subtract 1176 from both sides]

           f = 65                [divide both sides by (-2)]

There are 65 flies.

 

Check:

   Is  8(147-65) + 6*65 = 1046   ?

        8*82 + 6*65 = 1046   ?

          656 + 390 = 1046   ?

             1046 = 1046   ?yes

 

Answer 3

x = number of flies

y = number of spiders

 

6X = Fly legs

8Y = Spider legs

 

X + Y = 148

6X + 8Y = 1046

 

Since we want to know the number of flies, let's get rid of the Y value and substitute.

 

X + Y = 148

Y = 148 - X

 

6X + 8 (148 - X) = 1046

-2X + 1184 = 1046

-2X = -138

X = 69

 

To check...

69 + Y = 148

Y = 79

 

6 ( 69 ) + 8 (79) = 1046

414 + 632 = 1046

1046 + 1046

 

There are 69 flies

Answer 4

s represents the number of spiders

147 - s represents the number of flies

8s + 6(147 - s) = 1046

 

Can you solve for s?